Bernoulli / binomial trials for a process with variable probability of success

Question

One of the conditions for a Bernoulli trial (and by extension binomial proportion confidence intervals) is that the probability of success is the same every time the experiment is conducted.

In the case of a process that has some degree of unavoidable variability in it (e.g. p = 0.8 $\pm$ 0.1), is this still a Bernoulli trial (and can binomial proportion confidence intervals still be calculated)? I understand that if the outcome is yes / no, success / failure, then by definition it is a Bernoulli process, but I am unsure of how to deal with the changing probability of success. The mean stays the same between experiments, but there is variability around it - is that okay?

Context:

I'm looking at material breakage. My samples are similar sizes, but have slightly different shapes (like gravel).

Answer 1

You might be interested in the beta-bernoulli distribution. This is a compound distribution, where the probability of success is modeled as some beta distribution.

$$ p \sim \text{Beta}(b,a)\\ x \sim \text{Bernoulli}(p) $$

This seems appealing for your problem because the probability of success is explicitly modeled as random.